I tried patching the EU client. (I only have the Original WoW and TBC)

But anyway, I entered a random name and it started patching. But when it was just at the beginning it gave me an error and told me to close all applications and try again because it could not create the files.

I dont know what's happening...Have you tried the Technical Support forum ?
Or just go to the official WoW Forums and dig it to find if anyone has already got this problem.

Hmm, Doberman told me to download WOTLK, this might fix the issue.

Well, I've been downloading it for quite some while. It's only on 4% Is anyone sure that I have to have WOTLK? I doubt Blizzard would force one to buy their games.

I dont think wotlk is the issue , i think its the space in your Hard drive, might wanna check if you have enough space for the patches.